Abstract
The KKT optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts are proposed by defining an ordering relation on the class of all fuzzy numbers. Owing to this ordering relation being a partial ordering, the solution concepts proposed in this paper will follow from the similar solution concept, called Pareto optimal solution, in the conventional multiobjective programming problems. In order to consider the differentiation of fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the KKT optimality conditions are elicited naturally by introducing the Lagrange function multipliers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Apostol T.M. (1974). Mathematical analysis (2nd ed.). Addison-Wesley Publishing Company.
Banks H.T., Jacobs M.Q. (1970) A differential calculus for multifunctions. Journal of Mathematical Analysis and Applications 29: 246–272
Bazarra M.S., Sherali H.D., Shetty C.M. (1993) Nonlinear programming. Wiley, New York
Birge J.R., Louveaux F. (1997) Introduction to stochastic programming. Physica-Verlag, New York
Delgado M., Kacprzyk J., Verdegay J.-L., Vila M.A. (Eds.) (1994) Fuzzy optimization: Recent advances. Physica-Verlag, New York
Horst R., Pardalos P.M., Thoai N.V. (2000) Introduction to global optimization (2nd ed.). Kluwer Academic Publishers, Boston
Inuiguchi M., & Ramík J. (2000) Possibilistic linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets and Systems 111: 3–28
Kall P. (1976) Stochastic linear programming. Springer, New York
Lai Y.-J., Hwang C.-L. (1992) Fuzzy mathematical programming: Methods and applications, Lecture notes in economics and mathematical systems (Vol. 394). Springer, New York
Lai Y.-J., Hwang C.-L. (1994) Fuzzy Multiple objective decision making: Methods and applications, Lecture notes in economics and mathematical systems (Vol. 404). Springer, New York
Prékopa A. (1995) Stochastic programming. Kluwer Academic Publishers, Boston
Słowiński R. (Eds.) (1998) Fuzzy sets in decision analysis, operations research and statistics. Kluwer Academic Publishers, Boston
Słowiński R., Teghem J. (Eds.) (1990) Stochastic versus fuzzy approaches to multiobjective mathematical programming under uncertainty. Kluwer Academic Publishers, Boston
Stancu-Minasian, I. M. (1984). Stochastic programming with multiple objective functions. D. Reidel Publishing Company.
Vajda S. (1972) Probabilistic programming. Academic Press, New York
Wu H.-C. (2007) The Karush-Kuhn-Tucker optimality conditions for the optimization problem with fuzzy-valued objective function. Mathematical Methods of Operations Research 66: 203–224
Zadeh L.A. (1965) Fuzzy sets. Information and Control 8: 338–353
Zadeh, L. A. (1975). The concept of linguistic variable and its application to approximate reasoning I, II and III. Information Sciences, 8, 199–249, 301–357, 9, 43–80.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, HC. The Karush-Kuhn-Tucker optimality conditions for multi-objective programming problems with fuzzy-valued objective functions. Fuzzy Optim Decis Making 8, 1–28 (2009). https://doi.org/10.1007/s10700-009-9049-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-009-9049-2